6 190 D.A. SALAS-DE-LEÓN, M.A. MONREAL-GÓMEZ, E. AN-DE-EN, S. WEILAND, AND D. SALAS-MONREAL To acieve model reduction, we carry out a nonlinear Galerkin procedure wit te set of elements. How to coose te values of te nonlinear Galerkin transformation is a crucial question. Te associated POD eigenvalues sould define a relative information content to coose a low-dimensional basis by neglecting modes corresponding to te small eigenvalues in order to capture most of te energy of te snapsot basis. Here for our case, if te POD is constructed for 5 and a reduced order model wit 3 it yields a ratio of about 0.98; and if is constructed wit 15 it yields a ratio of above 0.99 for te percentage of kinetic energy retained (Fig. 5 and 6). We are now returning to te barotropic tidal and wind stress model for a costal lagoon in te Mexican Caribbean to apply te POD tecnique. Terefore, we solve Eqs. (4) - (6) after 5 tidal cycles of te M armonic. Results using classical model are depicted grapically in Figure 4. Te results of te model using POD are grapically almost te same and will not be sown. FIGRE 5. Ortogonal base and order evolution of te approximations. To quantify te performance of te reduced basis metod, we use two metrics namely te root mean square error (RMSE) and correlation of te difference between te full order and te reduced order simulation. Tis is obtained by first taking te five tidal cycles full order results and te corresponding five tidal cycles reduced order results and computing te error, for example, for te variable u and v components of te velocity vector ( v); te errors are sown in Figs. 6 and 7. Here, if n = 10 basis function, te first four PODs modes (Fig. 6), capture nearly 100%, wile for n = 15 basis function, te first seven POD (Fig. 6) capture nearly 100% wit an error ranging from 10 4 to Modes capture about 99% of te energy. Tus, different POD modes may be used to reconstruct fields respectively. For different numbers of snapsots but for te same energy percentage captured, te RMSE decrease stops at 15 snapsots. Te correlation taking te five tidal cycles full order results and te corresponding five tidal cycles reduced order clearly, wen increasing te POD mode, te correlation increases also for te same snapsots. Tis increase stops at 5 snapsots and te reported best approximation obtained wit 15 snapsots produced a correlation at te same level as te approximation 0 snapsots. However, one must also note tat a simple linear independence is not a sufficient criterion for coosing te POD mode. It only provides one wit some reference. Te error between te full order and te reduced order is displayed in Fig. 7 for a retained energy percentage of 99%. Tere is a little improvement between eiter 10 snapsots or 15 snapsots and 5 snapsots, but tere is almost no difference between 15 snapsots and 30 snapsots. Order approximation may be sufficiently close to te full order approximation. Oter experiments ave also been carried out, wit eiter more or fewer snapsots taken and for different percentages of energy captured, not sown ere. From te computational cost and memory storage aspects, 15 snapsots and te energy captured at 99% level yielded te best results. FIGRE 6. Computed error wit n = 15 basis functions. a) Absolute value of te currents, continuous line u, and dotted line v components of te velocity vector v, and b) absolute error of te current compared wit te classical barotropic ocean circulation numerical model results. FIGRE 7. Computed absolute error in v wit n= 15 base functions and K= 60 at time 140 for all positions in te numerical spatial grid. Rev. Mex. Fís. 55 (3) (009)

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